Communications in Mathematical Physics

, Volume 219, Issue 3, pp 631–669

Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory

  • Yasuyuki Kawahigashi
  • Roberto Longo
  • Michael Müger

DOI: 10.1007/PL00005565

Cite this article as:
Kawahigashi, Y., Longo, R. & Müger, M. Commun. Math. Phys. (2001) 219: 631. doi:10.1007/PL00005565

Abstract:

We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo–Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of ?. As a consequence, the index of ?(E)⊂?(E′)′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of ? form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Yasuyuki Kawahigashi
    • 1
  • Roberto Longo
    • 2
  • Michael Müger
    • 2
  1. 1.Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan.¶E-mail: yasuyuki@ms.u-tokyo.ac.jpJP
  2. 2.Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica,¶00133 Roma, Italy. E-mail: longo@mat.uniroma2.it; mueger@x4u.desy.deIT